# nLab twisted de Rham cohomology

Contents

### Context

#### Differential cohomology

differential cohomology

cohomology

# Contents

## Idea

Twisted de Rham cohomology is the twisted cohomology-version of de Rham cohomology, a simple example of twisted differential cohomology.

For degree-3 twists this is the codomain of the twisted Chern character on twisted K-theory, and in its orbifold cohomology-generalization it is the codomain of the twisted equivariant Chern character on twisted equivariant K-theory.

## Definition

### Plain

###### Definition

(1-twisted de Rham cohomology)
(…)

###### Definition

(3-twisted de Rham cohomology)
For $X$ a smooth manifold and $H \in \Omega^3(X)$ a closed differential 3-form, the $H$ twisted de Rham complex is the $\mathbb{Z}_2$-graded vector space $\Omega^{even}(X) \oplus \Omega^{odd}(X)$ equipped with the $H$-twisted de Rham differential

$d + H \wedge(-) \;\colon\; \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,$

Notice that this is nilpotent, due to the odd degree of $H$, such that $H \wedge H = 0$, and the closure of $H$, $d H = 0$.

###### Remark

There is also the cohomology of the chain complex whose maps are just multiplication by $H$.

$H \wedge(-) \;\colon\; \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,$

This is also called H-cohomology (Cavalcanti 03, p. 19).

### Equivariant

We discuss notions of twisted de Rham cohomology on (global quotient) orbifolds, as they are used for the codomain of the twisted equivariant Chern character on twisted equivariant K-theory.

This combines the above twistings in degrees 1 and 3, the latter induced from the curvature 3-form on a twisting 3-class, the former induced from the flat connection on the circle principal bundle (over the inertia orbifold) which is classified by the transgression of the 3-twist to a 2-class. This requires that this connection be flat, hence that the transgressed 2-class is torsion, which is guaranteed by a Lemma that we discuss first, in Transgression of the 3-twist to a 1-twist on Inertia.

In all of the the following:

• Let $G$ be a finite group.

• Let

(1)$X \in Proper G Actions(SmthMfds)$

be a proper smooth G-manifold, hence a smooth manifold equipped with a proper action (from the right, say) of $G$ by diffeomorphisms.

• Notice that for any $g \in G$ the fixed locus

(2)$X^g \xhookrightarrow{\;i_g\;} X$

is a smooth submanifold (by this Prop.).

All twisting classes, in the following are in ordinary Borel-equivariant cohomology, hence in the ordinary cohomology of a Borel construction. The “3-twist” and “torsion 2-twist” are in integral cohomology (of the Borel construction) and the “1-twist” has coefficients a finite cyclic group.

#### Transgression of 3-twist to 1-twist on inertia

We discuss the transgression of a 3-twist on a (global quotient) orbifold to a 1-twist on its inertia orbifold, namely to a torsion 2-class:

1. via the Künneth theorem

as argued in Becerra & Uribe 2009, Section 3.2;

2. via a Serre spectral sequence

as sketched in Freed, Hopkins & Teleman 07, (3.5).

There is also a corresponding argument in terms of bundle gerbes, given in Tu & Xu 2006, Prop. 2.6 & 3.6.

##### Via Künneth theorem

Throughout, fix an element $g \in G$.

Consider the right group action of the direct product group $C_G(g) \times \mathbb{Z}$ (of the centralizer subgroup with the integers) on the fixed locus $X^g$ (2) given by

(3)$\array{ X^g \times C_G(g) \times \mathbb{Z} &\xrightarrow{\;\;\;\;}& X^g \\ \big( x, \, (h, n) \big) &\mapsto& x \cdot h \cdot g^n \mathrlap{\,.} }$

and notice that the following function is a group homomorphism (by the fact that all elements of $C_G(g)$ commute with $g$ in $G$):

(4)$\array{ C_G(g) \times \mathbb{Z} &\xrightarrow{\;\;\phi_g\;\;}& G \\ (h,n) &\mapsto& h \cdot g^n \mathrlap{\,.} }$

In view of the action (3), the homomorphism (4) induces a map of Borel constructions

$\big( X^g \sslash C_G(g) \times B \mathbb{Z} \big) \;\simeq\; X^g \sslash \big( C_G(g) \times \mathbb{Z} \big) \xrightarrow{ \;\; i_g \sslash \phi_g \;\; } X \sslash G \,,$

(where the homotopy equivalence shown on the left follows since the group action of $\mathbb{Z}$ on $X^g$ is the trivial action, by definition (3), and using that the Borel construction on a point is the classifying space $\ast \sslash \mathbb{Z} \,\simeq\, B \mathbb{Z} \simeq S^1$, hence the circle, in the present case)

and hence the corresponding pullback in integral cohomology:

(5)$\begin{array}{rrl} H^\bullet \big( X \sslash G \; \, \mathbb{Z} \big) & \xrightarrow{ \; (i_g \sslash \phi_g)^\ast \; } & H^\bullet \big( ( X^g \sslash C_G(g) ) \times B \mathbb{Z} \; \, \mathbb{Z} \big) \\ & \,\simeq\, & H^\bullet \big( ( X^g \sslash C_G(g) ) \; \, \mathbb{Z} \big) \otimes_{\mathbb{Z}} \underset{ \mathclap{ \simeq \left\{ \array{ \mathbb{Z} &\vert& \bullet \in \{0,1\} \\ 0 &\vert& else } \right. } }{ \underbrace{ H^\bullet \big( B \mathbb{Z} ;\, \mathbb{Z} \big) } } \\ &\simeq & H^\bullet \big( ( X^g \sslash C_G(g) ) \; \, \mathbb{Z} \big) \,\oplus\, H^{\bullet-1} \big( ( X^g \sslash C_G(g) ) \; \, \mathbb{Z} \big) \,, \end{array}$

where the second line uses the Künneth theorem.

(The following definition becomes a proposition if one uses conceptualization of transgression as laid out in transgression in group cohomology.)

###### Definition

For $g \in G$ write

$\tau_g \;\colon\; H^\bullet \big( X \sslash G ;\, \mathbb{Z} \big) \xrightarrow{\;\;} H^{\bullet-1} \big( ( X^g \sslash C_G(g) ) ;\, \mathbb{Z} \big)$

for the composite of (5) with projection onto the second direct summand.

###### Proposition

The image of the transgression map $\tau_g$ (Def. ) is in the torsion subgroup.

(Becerra & Uribe 2009, Lemma 3.2)
###### Proof

The group homomorphism (4) factors through the cyclic group $\mathbb{Z} \to \langle g\rangle \to G$ which is generated by $g \in G$. By the assumption that $G$ is a finite group, so that its integral cohomology is entirely torsion. Since the transgression map factors through a tensor product with pullback along this factorization, by definition, the claim follows.

##### Via the Serre spectral sequence

###### Lemma

For each $g \in G$ and each point in the Borel construction $X^g \!\sslash\! \frac{C_G(g)}{\langle{g}\rangle}$, there is a homotopy fiber sequence of the form

(6)$\array{ B \langle{g}\rangle &\longrightarrow& X^g \!\sslash\! \langle{g}\rangle \\ \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ \ast &\longrightarrow& X^g \!\sslash\! \frac{C_G(g)}{\langle{g}\rangle} }$

###### Proof

First notice that the short exact sequence

$1 \to \langle{g}\rangle \hookrightarrow C_G(g) \twoheadrightarrow G/\langle{g}\rangle \to 1$

deloops to a homotopy fiber-sequence of the form

(7)$\array{ B \langle{g}\rangle &\longrightarrow& B C_G(g) \\ && \big\downarrow \\ && B \frac{C_G(g)}{\langle{g}\rangle} }$

(using this Prop.).

Then notice that we have a pasting diagram of homotopy pullbacks as follows:

$\array{ B \langle{g}\rangle &\longrightarrow& \ast \\ \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ X^g \!\sslash\! C_G(g) &\xrightarrow{\;}& X^g \!\sslash\! \frac{C_G(g)}{\langle{g}\rangle} \\ \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ B G &\xrightarrow{\;}& B \frac{C_G(g)}{\langle{g}\rangle} }$

Here:

• the bottom square being a homotopy pullback expresses that the group action of $\langle{g}\rangle$ on $X^g$ is trivial, so that the group action of $C_G(g)$ on $X^g$ is induced by that of $C_G(g)/\langle{g}\rangle$;

• the total rectangle is a homotopy pullback by (7);

• hence the top square is a homotopy pullback by the pasting law.

It follows – once we know (do we?) that the action of the fundamental group of $X^g \sslash \frac{C_G(g)}{\langle{g}\rangle}$ on the integral cohomology of $B\langle{g}\rangle$ is trivial – that we have a Serre spectral sequence

(8)$H^p \left( X^g \!\sslash\! \frac{C_G(g)}{\langle{g}\rangle} ;\, H^q \big( B\langle{g}\rangle{g} ;\, \mathbb{Z} \big) \right) \;\; \overset{\;\;\;\;}{\Rightarrow} \;\; H^{p+q} \big( X^g \!\sslash\! C_G(g) ;\, \mathbb{Z} \big) \,.$

But since $\langle{g}\rangle \simeq \mathbb{Z}/(ord(g))$ is a cyclic group (by the assumption that $G$ is a finite group) it follows (by this Prop. or Lem. 4.51 in BSST 07) that its group cohomology is concentrated in even degrees:

$H^n \big( B \langle{g}\rangle ;\, \mathbb{Z} \big) \;=\; H^n_{grp} \big( \langle{g}\rangle ;\, \mathbb{Z} \big) \;\simeq\; \left\{ \begin{array}{ccl} \mathbb{Z} &\vert& n = 0 \\ \mathbb{Z}/ord(g) &\vert& n = \,\text{positive multiple of}\, 2 \\ 0 &\vert& \text{otherwise} \end{array} \right.$

Therefore the spectral sequence (8) associates with any 3-twist $\alpha \mapsto \alpha_{\vert X^g} \in H^3\big( X^g \sslash C_G(g);\, \mathbb{Z} \big)$ a “transgressed” degree-1 class

(9)$\tau_g(\alpha) \;\in\; H^1 \left( X^g ;\, \mathbb{Z}/ord(g) \right) \xrightarrow{\;\;\beta\;\;} H^2 \left( X^g ;\, \mathbb{Z} \right) \,,$

which the Bockstein homomorphism identifies with a torsion 2-class in integral cohomology.

Essentially this conclusion is claimed as FHT 07, (3.5).

#### Definition

Now we can indicate the definition of the twisted equivariant de Rham cohomology. In outline:

Write $\Lambda (\prec (X \sslash G))$ for the inertia orbifold of the global quotient orbifold of $X$ (1).

For $\alpha \in H^3\big( X \sslash G; \, \mathbb{Z} \big)$ a “3-twist” in the degree-3 integral cohomology of the homotopy quotient (Borel construction) of $X$ by $G$ (1)

• let $H_3 \in \Omega^3\big( \Lambda (\prec (X \sslash G)) \big)$ be a de Rham image of $\alpha$ pulled back to the inertia orbifold,

• let $\nabla$ be a connection on the transgression (Def. ) of $\alpha$ to the inertia orbifold, which is flat by Prop. or, alternatively, by (9).

Then equivariant twisted de Rham cohomology of $X$ is the de Rham cohomology of $\Lambda (\prec (X \sslash G))$ which is both

• 1-twisted by $\nabla$ and

• 3-twisted by $H_3$

## Properties

###### Proposition

If the de Rham complex $(\Omega^\bullet(X),d_{dr})$ is formal, then for $H \in \Omega_{cl}^3(X)$ a closed differential 3-form, the $H$-twisted de Rham cohomology (Def. ) of $X$ coincides with its H-cohomology, for any closed 3-form $H$.

## References

### Twist in degree 1

The classical case of twisted de Rham cohomology with twists in degree 1, given by flat connections on flat line bundles and more generally on flat vector bundles), and its equivalence to sheaf cohomology with coefficients in abelian sheaves of flat sections (local systems):

Review:

• Cailan Li, Cohomology of Local Systems on $X_\Gamma$ (pdf, pdf)

• Youming Chen, Song Yang, Section 2.1 in: On the blow-up formula of twisted de Rham cohomology. Annals of Global Analysis and Geometry volume 56, pages 277–290 (2019) (arXiv:1810.09653, doi:10.1007/s10455-019-09667-8)

### Twist in degree 3

#### Plain

The concept of $H_3$-twisted de Rham cohomology was introduced (in discussion of the B-field in string theory), in:

Further discussion (often as the codomain of the twisted Chern character on twisted K-theory):

#### Equivariant

The generalization of 3-twisted de Rham cohomology to orbifolds (often as the codomain of the twisted equivariant Chern character on twisted equivariant K-theory):